Βασίλειος, Basilius=royal or kingly.  The name was a common one.  Fabricius catalogues “alii Basilii ultra xxx.,” all of some fame.  The derivation of Βασιλεύς is uncertain, and the connexion of the last syllable with λεύς=λέως=λαός, people, almost certainly wrong.  The root may be ÖBA, with the idea that the leader makes the followers march.  With the type of name, cf. Melchi and the compounds of Melech (e.g. Abimelech) in Scripture, and King, LeRoy, Koenig, among modern names.